If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied. For let the number A by multiplying the two numbers B, C make D, E; I say that, as B is to C, so is D to E. For, since A by multiplying B has made D, therefore B measures D according to the units in A. But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D. Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20] For the same reason, as the unit F is to the number A, so also is C to E; therefore also, as B is to D, so is C to E.